A volume comparison theorem for Finsler manifolds

Durán, Carlos

doi:10.1090/S0002-9939-98-04629-2

A volume comparison theorem for Finsler manifolds
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by Carlos E. Durán PDF

Proc. Amer. Math. Soc. 126 (1998), 3079-3082 Request permission

Abstract:

Let $(M^{n},F)$ be a symmetric Finsler manifold, endowed with the Busemann volume form, and let $D$ be its unit disk bundle endowed with the canonical symplectic volume form. It is shown that $Vol(D)\leq C(n)Vol(M^{n})$, where $C(n)$ is the volume of the unit disk in ${\mathbb {R}}^{n}$. Moreover, equality holds if and only if $(M^{n},F)$ is Riemannian.

References

H. Akbar-Zadeh, Sur les espaces de Finsler à courbures sectionnelles constantes, Acad. Roy. Belg. Bull. Cl. Sci. (5) 74 (1988), no. 10, 281–322 (French, with English summary). MR 1052466, DOI 10.3406/barb.1988.57782
D. Bao, S.-S. Chern and Z. Shen, An Introduction to Riemann-Finsler Geometry, Springer-Verlag, Berlin.
Robert L. Bryant, Finsler structures on the $2$-sphere satisfying $K=1$, Finsler geometry (Seattle, WA, 1995) Contemp. Math., vol. 196, Amer. Math. Soc., Providence, RI, 1996, pp. 27–41. MR 1403574, DOI 10.1090/conm/196/02427
A. R. Collar, On the reciprocation of certain matrices, Proc. Roy. Soc. Edinburgh 59 (1939), 195–206. MR 8, DOI 10.1017/S0370164600012281
Morgan Ward and R. P. Dilworth, The lattice theory of ova, Ann. of Math. (2) 40 (1939), 600–608. MR 11, DOI 10.2307/1968944
Saunders MacLane, Steinitz field towers for modular fields, Trans. Amer. Math. Soc. 46 (1939), 23–45. MR 17, DOI 10.1090/S0002-9947-1939-0000017-3
E. Cartan, Sur un problème d’equivalence et la théorie des espaces métriques generalisés, Ouvres Complétes, Vol. II, Part. III, Gauthier-Villars, Paris, pp. 1131-1153.
Mathieu Meyer and Alain Pajor, On Santaló’s inequality, Geometric aspects of functional analysis (1987–88), Lecture Notes in Math., vol. 1376, Springer, Berlin, 1989, pp. 261–263. MR 1008727, DOI 10.1007/BFb0090059
Z. Shen, On Finsler geometry of submanifolds, Preprint (1996).
A. C. Thompson, Minkowski geometry, Encyclopedia of Mathematics and its Applications, vol. 63, Cambridge University Press, Cambridge, 1996. MR 1406315, DOI 10.1017/CBO9781107325845